A decreasing step method for strongly oscillating stochastic models

TL;DR

This paper introduces a new algorithm combining Euler schemes and decreasing step estimators to efficiently approximate solutions of strongly oscillating stochastic differential equations, with proven convergence and improved complexity.

Contribution

It presents a novel decreasing step method for strongly oscillating SDEs, leveraging homogenization and providing convergence and limit results.

Findings

Proves strong convergence of the proposed algorithm.

Establishes a CLT-like limit for the normalized error.

Introduces an extrapolated version with lower complexity.

Abstract

We propose an algorithm for approximating the solution of a strongly oscillating SDE, that is, a system in which some ergodic state variables evolve quickly with respect to the other variables. The algorithm profits from homogenization results and consists of an Euler scheme for the slow scale variables coupled with a decreasing step estimator for the ergodic averages of the quick variables. We prove the strong convergence of the algorithm as well as a C.L.T. like limit result for the normalized error distribution. In addition, we propose an extrapolated version that has an asymptotically lower complexity and satisfies the same properties as the original version.